The techniques we use to study (REF ) can also be applied to the corresponding Stefan problem (REF ) to conclude interior Hölder continuity for \(v=\phi (u)\) .
Then, Hölder continuity of \(u\)  can be inferred provided that \(\beta :=\phi ^{-1}\)  is Hölder continuous.
This was the original method applied to the porous medium equation, see [1]}.
In fact, this is the natural approach, since a typical property of Stefan problems is that their solutions \(v\)  enjoy better continuity properties than \(u=\beta (v)\) , see [2]}.
Therefore, if Hölder continuity for \(\phi (u)\)  cannot be deduced, then it is unlikely that Hölder continuity for \(u\)  can be obtained.
This can be made rigorous if \(\phi \)  and \(\beta :=\phi ^{-1}\)  are both Hölder continuous.
However, in our case \(\phi \)  is not Hölder continuous due to the singularity \(\phi (1)=\infty \) .
We remark that we chose to study the original equation (REF ), since it describes the quantity of interest in applications.
